A Multi-Criteria Decision-Making Model for Pontoon Bridge Selection: An Application of the DIBR II-NWBM-FF MAIRCA Approach

Historically, as organized human communities manifested an intrinsic desire to expand territories, the imperative to overcome natural water barriers arose. These barriers, formidable in their essence, demanded innovation from these communities: initial solutions were found in swimming, which later evolved to the construction of varied watercraft. Coinciding with the inception of rudimentary military units, specialized brigades dedicated to circumventing these water impediments were established, equipped with intricate apparatuses tailored for the task [1]. Present-day military operations worldwide acknowledge the gravity of this combat action, which, given its intricate nuances, is often regarded as one of the most formidable combat challenges [1]. Furthermore, many military entities, beyond amphibious capabilities, are endowed with pontoon parks, encompassing an array of technical assets such as pontoons, vehicles, watercraft, and auxiliary tools, conceptualized for scaffold and bridge assembly [1], [2].

For decision-makers, the procurement of such assets designed to overcome water impediments presents a multifaceted challenge. The complexity arises from the plethora of dimensions or criteria by which these means are evaluated. Effective decision-making in this domain necessitates reliance on intricate tools for decision support, especially those adept at handling uncertainty and imprecision. Given the multidimensional nature of the problem at hand, the Multi-Criteria Decision-Making (MCDM) methods emerge as the apt tools for informed decision-making. These methods incorporate a variety of mathematical models and tools within the ambit of MCDM.

Typically, the resolution of an MCDM problem can be delineated into distinct steps: 1) Criteria identification pivotal to decision-making; 2) Ascertainment of criteria weight coefficients, denoting the influence exerted by each criterion on the final judgement; 3) Establishment of the initial decision-making matrix; 4) Employment of mathematical strategies for discerning the optimal solution; and 5) Selection of said optimal solution. To date, an extensive array of methods have been introduced. Some cater to determining criteria weight coefficients [3], [4], [5], [6], [7], [8], while others focus on the discernment of the optimal choice amongst a myriad of possibilities [3], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. The application of MCDM methodologies across diverse sectors has been documented extensively in previous research, as elucidated in Table 1.

In this paper, the DIBR II-NWBM-FF MAIRCA MCDM model (Figure 1) is presented for the selection of the optimal set of pontoon bridge for the needs of the military.

The model consists of the following stages: 1) Identification of criteria and their weight coefficients using the DIBR II method; 2) Choosing the optimal alternative using the FF MAIRCA method; 3) Sensitivity analysis of the output results of the method; and 4) Comparative analysis.

In alignment with the phases delineated in Figure 1, the subsequent sections elucidate the methods integrated within the MCDM model. Additionally, the I operators, pivotal for the aggregation of expert opinions, are detailed.

Introduced in 2023, the DIBR II method for determining weight coefficients of criteria is expounded in the study [8]. This method was devised to address the limitations observed in prior methodologies for calculating criteria weight coefficients. To date, its implementation has been documented in two distinct studies. In their research, Božanić and Pamučar [8] deployed this method in the domains of car procurement and the evaluation of social media efficacy. Subsequently, Božanić et al. [57] harnessed the method to determine weight coefficients of criteria, specifically while categorising methods and techniques within Lean organisation systems management.

A concise mathematical representation of the DIBR II method is presented below [8].

Step 1. Identification of criteria $K=\left\{ {{K}_{1}},{{K}_{2}},...,{{K}_{n}} \right\}$.

Step 2. Defining the rank of criteria based on importance ${{K}_{1}}>{{K}_{2}}>...>{{K}_{n}}$.

Step 3. Defining the relationship between the criteria.

...

Step 4. Defining the relationship between the most important and other criteria.

...

Step 5. Determination of the weight coefficient of the most important criterion.

Step 6. Determination of the weight coefficients of the remaining criteria.

Step 7. Evaluation of the quality of defined relationships.

In order to determine the quality of the defined relationships, it is necessary that the deviation ${{D}_{n}}$ of the criteria ${{K}_{n}}$ is in the range $0\le {{D}_{n}}\le 0.1$, where:

and where the values should be approximately equal, that is, a deviation of up to 10% is allowed.

Within the context of this investigation, enhancements to the MAIRCA method [18] were made, incorporating FRFSs. Detailed explications of these sets can be found in studies [58], [59], [60]. The mathematical architecture of the FF MAIRCA method, as illustrated in studies [18], [59], is delineated as follows:

Step 1. A linguistic scale was defined for the evaluation of alternatives, grounded on FRFSs.

Step 2. Every alternative $A=\left\{ {{A}_{1}},{{A}_{2}},...,{{A}_{i}} \right\}$ was evaluated by each expert $E=\left\{ {{E}_{1}},{{E}_{2}},...,{{E}_{k}} \right\}$ according to all criteria $C=\left\{ {{C}_{1}},{{C}_{2}},...,{{C}_{j}} \right\}$ using the established linguistic scale. Subsequently, the initial decision matrix $\otimes {{E}_{ijk}}=\left\{ {{\varphi }_{{{E}_{ijk}}}},{{\gamma }_{{{E}_{ijk}}}} \right\}$ was derived for each expert, wherein ${{\varphi }_{{{E}_{ijk}}}}$ represents the degree of membership,${{\gamma }_{{{E}_{ijk}}}}$ represents the degree of non-membership of FF number $\otimes {{E}_{ijk}}$, and $0\le {{\left( {{\varphi }_{(x)}} \right)}^{3}}+{{\left( {{\gamma }_{(x)}} \right)}^{3}}\le 1$.

Step 3. Expert opinions were aggregated and the initial decision matrix was attained, accomplished through the application of Eq. (11) [59].

where, $p$ represents the number of experts.

Step 4. The initial decision matrix $\otimes {{N}_{ij}}$ was normalized, achieved using Eq. (12) [59].

Step 5. The probability of opting for specific alternatives $P_{A_i}$ was discerned through Eq. (13) [18].

where, $i$ represents the total number of alternatives.

Step 6. The matrix of theoretical weights $\otimes {{T}_{pj}}$ was determined Eq. (14).

Step 7. The matrix of real weights $\otimes {{T}_{rij}}$ was determined Eq. (15).

where, $\otimes t_{rij} = \omega_{j} \cdot \otimes N_{ij}$.

Step 8. A gap matrix $\otimes {{G}_{ij}}$ was computed, contrasting theoretical and real weights, based on Eq. (16).

Step 9. The expected solution $\otimes {{Q}_{i}}$ was extrapolated via Eq. (17).

Step 10. Alternatives were sorted in accordance with the positive score function ${{\psi }^{p}}\left( \otimes {{Q}_{i}} \right)$ Eq. (17) [59] of the expected solution. It is paramount to note that the least score function value designates the top-rated alternative, and the converse holds true.

Step 11. The dominance index for the foremost-ranked alternative ${{I}_{D,\,1-i}}$ was discerned utilizing Eq. (19) and the dominance threshold ${{T}_{D}}$ through Eq. (20).

where, ${{Q}_{fr}}$ represents the first-ranked alternative, ${{Q}_{lr}}$ represents the last-ranked alternative and ${{Q}_{i}}$ represents the alternative being considered.

where, $i$ represents the total number of alternatives.

For the solution to be considered valid, the relationship must satisfy ${{I}_{D,\,1-i}}\ge {{T}_{D}}$. If not met, the alternative's rank must be amended. Such an alternative will bear a mark (1*) adjacent to its rank, indicating that while it is not the foremost choice, it remains a potential optimal solution. This implies that decision-makers might still consider it as a viable resolution to the MCDM problem.

This operator was adopted in the study to aggregate the opinions of seven experts when determining the weight coefficients of criteria using the DIBR II method Eq. (21).

where, $y_1, y_2, \ldots, y_n$ represent a set of positive numbers, $p, q \geq 0$ are stabilization parameters of the function, and $\omega_{i j}$ denote weight coefficients of experts' competencies.

The first step of the proposed methodology entailed criteria identification. Through a comprehensive analysis of extant literature and consultations with seven field experts $E=\left\{E_1, E_2, \ldots, E_7\right\}$, five distinct criteria $C=\left\{C_1, \mathrm{C}_2, \mathrm{C}_3, \mathrm{C}_4\right\}$ were established, as detailed in Table 2.

Subsequent to the criteria delineation, the DIBR II method was employed to define these criteria further. Seven experts from the domain determined the significance of each criterion, yielding respective weighting coefficients $\omega _{k}^{E}=\left( 0.155,\,0.135,\,0.140,\,0.130,\,0.145,\,0.135,\,0.160 \right)$. Eqs. (1)-(10) were employed to compute the weight coefficients of the criteria for every expert, as displayed in Table 3.

Opinions from experts were aggregated through the NWBM aggregator, specifically using Eq. (21). This aggregation process produced the final values of the weight coefficients of the criteria shaping the subject choice, as elucidated in Table 4.

Following the ascertainment of the weight coefficients of criteria, alternatives were identified. These alternatives were manifested as four distinct pontoon bridge sets available in the market, each possessing unique attributes $A=\left\{ {{A}_{1}},{{A}_{2}},{{A}_{3}},{{A}_{4}} \right\}$.

Step 1. Respecting the steps of the proposed FF MAIRCA methodology, Table 5 shows the linguistic scale employed by experts to evaluate each alternative against all criteria.

Step 2. Seven experts evaluated all alternatives using a linguistic scale (Table 5), and their decision matrices are given in Table 6.

Step 3. By means of Eq. (11), the aggregation of expert opinions was carried out, which leads to the initial decision-making matrix (Table 7).

Step 4. By applying Eq. (12), the initial decision matrix was normalized, and the normalized values are presented in Table 8.

Step 5. The probability of choosing certain alternatives was determined using Eq. (13) and the value 0.25 was obtained.

Step 6. Applying Eq. (14) leads to the matrix of theoretical weights (Table 9).

Step 7. The matrix of real weights was obtained by applying Eq. (15) and presented in Table 10.

Step 8. The matrix of the gap between theoretical and real weights obtained by applying Eq. (16) is presented in Table 11.

Step 9. The expected solutions are calculated using Eq. (17), and the obtained values are shown in Table 12.

Step 10. Applying Eq. (18) leads to the score function based on which the ranking is made (Table 13).

Step 11. By applying Eqs. (19) and (20), the dominance index for top-ranked alternative $I_{D, 1-i}$ and the threshold of dominance $T_D$ are determined ( Table 14).

Drawing insights from Table 14, it was concluded that the preliminary ranking coincided with the final ranking of alternatives. Specifically, alternative $A_4$ emerged as the top-ranked choice, while alternative $A_3$ was positioned as the least preferred option.

A sensitivity analysis concerning the introduced methodology was undertaken, focusing primarily on variations in the weight coefficients of the criteria, as delineated in studies [22], [24], [26], [28]. Within this analytical framework, 14 distinct scenarios of weight change were articulated, as illustrated in Figure 2.

By applying the scenarios shown in Figure 2, the following ranks of alternatives were obtained (see Figure 3).

Drawn from the changes in the rankings of the alternatives, as presented in Figure 4, it was inferred that the proposed FF MAIRCA method exhibited pronounced sensitivity to shifts in the weight coefficients of the criteria. Hence, when these coefficients are defined by experts, heightened vigilance is necessitated to discern the interrelationships among the criteria influencing the selection of the optimal alternative.

A comparative analysis ensued, juxtaposing the outcomes derived from the proposed methodology with those presented in studies [58], [60]. Employing the initial decision matrix (Table 7) and invoking mathematical constructs such as FFWPA (Fermatean Fuzzy weighted power average), FFWPG (Fermatean Fuzzy weighted power geometric), FFWA (Fermatean Fuzzy weighted average), and FFWG (Fermatean Fuzzy weighted geometric) operators, rankings of alternatives were formulated, as enumerated in Table 15.

The rankings acquired using the FF MAIRCA method and those presented in Table 15 were subsequently analyzed through the computation of the Spearman’s correlation coefficient, as discussed in studies [26], [62] (see Figure 4).

The data visualized in Figure 4 underscored the robustness of the proposed model. Specifically, the correlation coefficients displayed a proclivity towards a positive ideal correlation, suggesting the model's stability and its capacity to yield accurate results.

Through the application of the intricate MCDM model, encompassing DIBR II - NWBM - FF MAIRCA, a selection process for pontoon bridge sets was undertaken. The findings and insights derived from this study shed light on the nuances of decision-making across varied contexts. A pivotal aspect of the research involved leveraging the DIBR II method to ascertain weight coefficients of the criteria. This element was instrumental in delineating priorities among diverse criteria, grounded in the aggregated perspectives of seven experts via the NWBM operator. Subsequently, the weight coefficients were integrated into the FF MAIRCA method, showcasing an innovative enhancement of the MAIRCA approach utilizing FF sets. Serving as the foundation for alternative ranking, this method also facilitated the identification of the optimal solution for the MCDM quandary. The findings pinpointed alternative $A_4$ as the prime solution for the MCDM challenge associated with pontoon bridge set selection.

Further scrutiny was directed towards the sensitivity of the model relative to shifts in the weight coefficients of the criteria. It was inferred that pronounced sensitivity was exhibited, underscoring the imperative of meticulous weight definition by experts to maintain model precision in practical scenarios. For validation, a comparative analysis juxtaposed the results with the FFW operator's outcomes. Conclusions drawn verified the stability and accuracy of the proposed methodology, reinforcing its efficacy and applicability within the pontoon bridge set selection milieu.

This study furnishes a robust framework tailored for decisions concerning water obstacle traversal tools, with an emphasis on pontoon bridge sets. However, it is salient to highlight the necessity for rigorous criteria weight delineation to bolster the model's stability and accuracy. The devised methodology stands as a potential asset for decision-makers overcoming water obstacle challenges in military settings, aiming to bolster the selection efficacy of military resources.

The primary constraint of this investigation pivots on the quantity of criteria steering the choices. Future research trajectories allude to this limitation, advocating a granular exploration of criteria, coupled with the integration of theories adept at handling vagueness and uncertainties inherent in conventional MCDM methodologies.